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Existence of Positive and Sign-Changing Solutions to a Coupled Elliptic System with Mixed Nonlinearity Growth

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Abstract

In the present paper, we make a rigorous study of the solitary wave solutions to a coupled Schrödinger system with quadratic and cubic nonlinearity. This kind of system of Schrödinger equations arises from optics theory. First, the existence and nonexistence of nontrivial solutions, respectively, in focusing and defocusing cases are considered. Second, we prove the existence of multiple nontrivial solutions by using the Crandall–Rabinowitz local bifurcation theorems and calculate the exact Morse index of these solutions. Third, the continuous dependence on the parameter and asymptotic behavior of positive ground state solutions in the focusing case are also established. Particularly, from the mathematical point of view, we prove the behavior of positive solution coincides with the physical phenomena of Bang et al. (Opt Lett 22(22):1680–1682, 1997; Phys Rev E (3) 58(4):5057–5069, 1998). Finally, we prove the existence of sign-changing solutions.

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References

  1. Alexander, J.C., Antman, S.S.: Global and local behavior of bifurcating multidimensional continua of solutions for multiparameter nonlinear eigenvalue problems. Arch. Ration. Mech. Anal. 76(4), 339–354 (1981)

    MathSciNet  MATH  Google Scholar 

  2. Ambrosetti, A., Colorado, E.: Bound and ground states of coupled nonlinear Schrödinger equations. C. R. Math. Acad. Sci. Paris 342(7), 453–458 (2006)

    MathSciNet  MATH  Google Scholar 

  3. Ambrosetti, A., Colorado, E.: Standing waves of some coupled nonlinear Schrödinger equations. J. Lond. Math. Soc. (2) 75(1), 67–82 (2007)

    MathSciNet  MATH  Google Scholar 

  4. Bang, O.: Dynamical equations for wave packets in materials with both quadratic and cubic response. J. Opt. Soc. Am. B 14(52), 1677 (1997)

    Google Scholar 

  5. Bang, O., Bergé, L., Rasmussen, J.J.: Wave collapse in bulk media with quadratic and cubic responses. Opt. Commun. 146(1–6), 231–235 (1998)

    ADS  Google Scholar 

  6. Bang, O., Kivshar, S.-Y., Buryak, A.-V.: Bright spatial solitons in defocusing kerr media supported by cascaded nonlinearities. Opt. Lett. 22(22), 1680–1682 (1997)

    ADS  Google Scholar 

  7. Bang, O., Kivshar, Y.-S., Buryak, A.-V., Rossi, A.-D., Trillo, S.: Two-dimensional solitary waves in media with quadratic and cubic nonlinearity. Phys. Rev. E (3) 58(4), 5057–5069 (1998)

    ADS  MathSciNet  Google Scholar 

  8. Bartsch, T.: Bifurcation in a multicomponent system of nonlinear Schrödinger equations. J. fixed point theory appl. 13(1), 37–50 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Bartsch, T., Dancer, N., Wang, Z.-Q.: A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system. Calc. Var. Part. Differ. Equ. 37(3–4), 345–361 (2010)

    MathSciNet  MATH  Google Scholar 

  10. Bartsch, T., Liu, Z.-L.: On a superlinear elliptic \(p\)-Laplacian equation. J. Differ. Equ. 198(1), 149–175 (2004)

    ADS  MathSciNet  MATH  Google Scholar 

  11. Bartsch, T., Liu, Z.-L., Weth, T.: Sign changing solutions of superlinear Schrödinger equations. Commun. Part. Differ. Equ. 29(1–2), 25–42 (2004)

    MATH  Google Scholar 

  12. Bartsch, T., Liu, Z.-L., Weth, T.: Nodal solutions of a \(p\)-Laplacian equation. Proc. Lond. Math. Soc. (3) 91(1), 129–152 (2005)

    MathSciNet  MATH  Google Scholar 

  13. Bartsch, T., Wang, Z.-Q.: Note on ground states of nonlinear Schrödinger systems. J. Part. Differ. Equ. 19(3), 200–207 (2006)

    MATH  Google Scholar 

  14. Bartsch, T., Wang, Z.-Q., Wei, J.-C.: Bound states for a coupled Schrödinger system. J. fixed point theory appl. 2(2), 353–367 (2007)

    MathSciNet  MATH  Google Scholar 

  15. Bartsch, T., Dancer, N., Wang, Z.-Q.: A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system. Calc. Var. Part. Differ. Equ. 37(3–4), 345–361 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Bates, P.-W., Shi, J.-J.: Existence and instability of spike layer solutions to singular perturbation problems. J. Funct. Anal. 196(2), 211–264 (2002)

    MathSciNet  MATH  Google Scholar 

  17. Bergé, L., Bang, O., Rasmussen, J.-J., Mezentsev, V.-K.: Self-focusing and solitonlike structures in materials with competing quadratic and cubic nonlinearities. Phys. Rev. E 55(3), 3555–3570 (1997)

    ADS  MathSciNet  Google Scholar 

  18. Brézis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl. Math. 36(4), 437–477 (1983)

    MathSciNet  MATH  Google Scholar 

  19. Busca, J., Sirakov, B.: Symmetry results for semilinear elliptic systems in the whole space. J. Differ. Equ. 163(1), 41–56 (2000)

    ADS  MathSciNet  MATH  Google Scholar 

  20. Busca, J., Sirakov, B.: Symmetry results for semilinear elliptic systems in the whole space. J. Differ. Equ. 163(1), 41–56 (2000)

    ADS  MathSciNet  MATH  Google Scholar 

  21. Chang, S.-M., Lin, C.-S., Lin, T.-C., Lin, W.-W.: Segregated nodal domains of two-dimensional multispecies Bose–Einstein condensates. Phys. D 196(3–4), 341–361 (2004)

    MathSciNet  MATH  Google Scholar 

  22. Chen, Z.-J., Lin, C.-S., Zou, W.-M.: Infinitely many sign-changing and semi-nodal solutions for a nonlinear Schrodinger system (2012). arXiv:1212.3773

  23. Chen, Z.-J., Lin, C.-S., Zou, W.-M.: Multiple sign-changing and semi-nodal solutions for coupled Schrödinger equations. J. Differ. Equ. 255(11), 4289–4311 (2013)

    ADS  MATH  Google Scholar 

  24. Chen, Z.-J., Lin, C.-S., Zou, W.-M.: Sign-changing solutions and phase separation for an elliptic system with critical exponent. Commun. Part. Differ. Equ. 39(10), 1827–1859 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Chen, Z.-J., Zou, W.-M.: Positive least energy solutions and phase separation for coupled Schrödinger equations with critical exponent. Arch. Ration. Mech. Anal. 205(2), 515–551 (2012)

    MathSciNet  MATH  Google Scholar 

  26. Crandall, M.-G., Rabinowitz, P.-H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)

    MathSciNet  MATH  Google Scholar 

  27. Crandall, M.-G., Rabinowitz, P.-H.: Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Ration. Mech. Anal. 52, 161–180 (1973)

    MathSciNet  MATH  Google Scholar 

  28. Dancer, E.N., Wei, J.-C., Weth, T.: A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(3), 953–969 (2010)

    ADS  MathSciNet  MATH  Google Scholar 

  29. Dancer, E.N., Wei, J.: Spike solutions in coupled nonlinear Schrödinger equations with attractive interaction. Trans. Am. Math. Soc. 361(3), 1189–1208 (2009)

    MATH  Google Scholar 

  30. de Figueiredo, D.G., Lopes, O.: Solitary waves for some nonlinear Schrödinger systems. Ann. Inst. H. Poincaré Anal. Non Linéaire 25(1), 149–161 (2008)

    ADS  MathSciNet  MATH  Google Scholar 

  31. Du, Y.-H., Shi, J.-P.: Allee effect and bistability in a spatially heterogeneous predator-prey model. Trans. Am. Math. Soc. 359(9), 4557–4593 (2007)

    MathSciNet  MATH  Google Scholar 

  32. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    MathSciNet  MATH  Google Scholar 

  33. Esry, B., Greene, C., Burke, J., Bohn, J.: Hartree–Fock theory for double condensates. Phys. Rev. Lett. 78, 3594–3597 (1997)

    ADS  Google Scholar 

  34. Ikoma, N.: Uniqueness of positive solutions for a nonlinear elliptic system. NoDEA Nonlinear Differ. Equ. Appl. 16(5), 555–567 (2009)

    MathSciNet  MATH  Google Scholar 

  35. Kwong, M.K.: Uniqueness of positive solutions of \(\Delta u-u+u^p=0\) in \({ R}^n\). Arch. Ration. Mech. Anal. 105(3), 243–266 (1989)

    MATH  Google Scholar 

  36. Lin, L., Liu, Z., Chen, S.: Multi-bump solutions for a semilinear Schrödinger equation. Indiana Univ. Math. J. 58(4), 1659–1689 (2009)

    MathSciNet  MATH  Google Scholar 

  37. Lin, T.C., Wei, J.C.: Ground state of \(N\) coupled nonlinear Schrödinger equations in \({\mathbb{R}}^n\), \(n\le 3\). Commun. Math. Phys. 255(3), 629–653 (2005)

    ADS  MATH  Google Scholar 

  38. Lin, T.C., Wei, J.C.: Spikes in two-component systems of nonlinear Schrödinger equations with trapping potentials. J. Differ. Equ. 229(2), 538–569 (2006)

    ADS  MATH  Google Scholar 

  39. Lin, T.-C., Wei, J.-C.: Erratum: “Ground state of \(N\) coupled nonlinear Schrödinger equations in \({{ R}}^n\), \(n\le 3\)” [Comm. Math. Phys. 255 (2005), no. 3, 629–653; mr2135447]. Commun. Math. Phys. 277(2), 573–576 (2008)

    ADS  Google Scholar 

  40. Liu, J.-Q., Liu, X.-Q., Wang, Z.-Q.: Multiple mixed states of nodal solutions for nonlinear Schrödinger systems. Calc. Var. Part. Differ. Equ. 52(3–4), 565–586 (2015)

    MATH  Google Scholar 

  41. Liu, Z.-L., Wang, Z.-Q.: Multiple bound states of nonlinear Schrödinger systems. Commun. Math. Phys. 282(3), 721–731 (2008)

    ADS  MATH  Google Scholar 

  42. Liu, Z.-L., Wang, Z.-Q., Zhang, J.-J.: Infinitely many sign-changing solutions for the nonlinear Schrödinger–Poisson system. Accepted by Annali di Matematica Pura ed Applicata (2014)

  43. Lopes, O.: Uniqueness of a symmetric positive solution to an ODE system. Electron. J. Differ. Equ. 162, 8 (2009)

    MathSciNet  MATH  Google Scholar 

  44. Maia, L.A., Montefusco, E., Pellacci, B.: Positive solutions for a weakly coupled nonlinear Schrödinger system. J. Differ. Equ. 229(2), 743–767 (2006)

    ADS  MATH  Google Scholar 

  45. Maia, L.A., Montefusco, E., Pellacci, B.: Infinitely many nodal solutions for a weakly coupled nonlinear Schrödinger system. Commun. Contemp. Math. 10(5), 651–669 (2008)

    MathSciNet  MATH  Google Scholar 

  46. Mitchell, M., Segev, M.: Self-trapping of incoherent white light. Nature 387(6636), 880–883 (1997)

    ADS  Google Scholar 

  47. Ni, W.-M., Takagi, I.: Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J. 70(2), 247–281 (1993)

    MathSciNet  MATH  Google Scholar 

  48. Pomponio, A.: Ground states for a system of nonlinear Schrödinger equations with three wave interaction. J. Math. Phys. 51(9), 093513, 20 (2010)

    MathSciNet  MATH  Google Scholar 

  49. Pucci, P., Serrin, J.: A general variational identity. Indiana Univ. Math. J. 35(3), 681–703 (1986)

    MathSciNet  MATH  Google Scholar 

  50. Rossi, A.-D., Assanto, G., Trillo, S., Torruellas, S.W.-E.: Excitation of stable transverse wavepackets with quadratic and cubic susceptibilities. Opt. Commun. 150(1–6), 390–398 (1998)

    ADS  Google Scholar 

  51. Shi, J.-P.: Persistence and bifurcation of degenerate solutions. J. Funct. Anal. 169(2), 494–531 (1999)

    MathSciNet  MATH  Google Scholar 

  52. Sirakov, B.: Least energy solitary waves for a system of nonlinear Schrödinger equations in \({\mathbb{R}}^n\). Commun. Math. Phys. 271(1), 199–221 (2007)

    ADS  MATH  Google Scholar 

  53. Struwe, M.: Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, vol. 34, 4th edn. Springer, Berlin (2008)

    MATH  Google Scholar 

  54. Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 4(110), 353–372 (1976)

    MathSciNet  MATH  Google Scholar 

  55. Timmermans, E.: Phase separation of Bose–Einstein condensates. Phys. Rev. Lett. 81, 5718–5721 (1998)

    ADS  Google Scholar 

  56. Wang, J.: Solitary waves for coupled nonlinear elliptic system with nonhomogeneous nonlinearities. Calc. Var. Partial Differ. Equ. 56(2), 38 (2017)

    MathSciNet  MATH  Google Scholar 

  57. Wang, J., Shi, J.-P.: Standing waves of a weakly coupled Schrödinger system with distinct potential functions. J. Differ. Equ. 260(2), 1830–1864 (2016)

    ADS  MATH  Google Scholar 

  58. Wang, J., Shi, J.-P.: Standing waves of coupled Schrodinger equations with quadratic interactions from Raman amplification in a plasma. Preprint (2017)

  59. Wang, J., Tian, L.-X., Xu, J.-X., Zhang, F.-B.: Existence and nonexistence of the ground state solutions for nonlinear Schrödinger equations with nonperiodic nonlinearities. Math. Nachr. 285(11–12), 1543–1562 (2012)

    MathSciNet  MATH  Google Scholar 

  60. Wang, J., Tian, L.-X., Xu, J.-X., Zhang, F.-B.: Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth. J. Differ. Equ. 253(7), 2314–2351 (2012)

    ADS  MathSciNet  MATH  Google Scholar 

  61. Wei, J.-C., Weth, T.: Radial solutions and phase separation in a system of two coupled Schrödinger equations. Arch. Ration. Mech. Anal. 190(1), 83–106 (2008)

    MathSciNet  MATH  Google Scholar 

  62. Wei, J.-C., Winter, M.: Critical threshold and stability of cluster solutions for large reaction–diffusion systems in \({\mathbb{R}}^1\). SIAM J. Math. Anal. 33(5), 1058–1089 (2002). (electronic)

    MathSciNet  MATH  Google Scholar 

  63. Willem, M.: Minimax Theorems. Progress in Nonlinear Differential Equations and Their Applications, vol. 24. Birkhäuser Boston Inc, Boston (1996)

    Google Scholar 

  64. Yew, A.C.: Multipulses of nonlinearly coupled Schrödinger equations. J. Differ. Equ. 173(1), 92–137 (2001)

    ADS  MathSciNet  MATH  Google Scholar 

  65. Yew, A.C., Champneys, A.R., McKenna, P.J.: Multiple solitary waves due to second-harmonic generation in quadratic media. J. Nonlinear Sci. 9(1), 33–52 (1999)

    ADS  MathSciNet  MATH  Google Scholar 

  66. Zhao, L.-G., Zhao, F.-K., Shi, J.-P.: Higher dimensional solitary waves generated by second-harmonic generation in quadratic media. Calc. Var. Part. Differ. Equ. 54(3), 2657–2691 (2015)

    MathSciNet  MATH  Google Scholar 

  67. Zou, W.-M.: Sign-Changing Critical Point Theory. Springer, New York (2008)

    MATH  Google Scholar 

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Acknowledgements

The authors thank the referee’s thoughtful reading of details of the paper and nice suggestions to improve the results. This work was supported by NNSFC (Grants 11971202, 11671077) and the Six big talent peaks project in Jiangsu Province (XYDXX-015).

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Authors and Affiliations

  1. School of Mathematical Science, Jiangsu University, Zhenjiang, 212013, People’s Republic of China

    Jun Wang

  2. Institute of Applied System Analysis, Jiangsu University, Zhenjiang, 212013, People’s Republic of China

    Jun Wang

  3. Department of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Junxiang Xu

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  1. Jun Wang
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Correspondence to Jun Wang.

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Wang, J., Xu, J. Existence of Positive and Sign-Changing Solutions to a Coupled Elliptic System with Mixed Nonlinearity Growth. Ann. Henri Poincaré 21, 2815–2860 (2020). https://doi.org/10.1007/s00023-020-00937-x

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